Lattice Points and Simultaneous Core Partitions

We observe that for a and b relatively prime, the “abacus construction” identifies the set of simultaneous (a, b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a, b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson’s theorem [3] that there are (a + b− 1)!/a!b! simultaneous (a, b)-cores, and using Euler-Maclaurin theory we prove Armstrong’s conjecture [12] that the average size of an (a, b)core is (a + b + 1)(a− 1)(b− 1)/24. Our methods also give new derivations of analagous formulas for the number and average size of self-conjugate (a, b)-cores. We conjecture a unimodality result for q rational Catalan numbers, and make preliminary investigations in applying these methods to the (q, t)-symmetry and specialization conjectures. We prove these conjectures for low degree terms and when a = 3, connecting them to the Catalan hyperplane arrangement and proving an apparently new result about permutation statistics along the way.